The final か in the VかVか grammar is actually more related to the か grammar point "Creates a question within a sentence" than being a part of the "Or" grammar point; when you're presenting the option between 2 verbs you're essentially asking the full question "Will you do V?" (because a verb can be a full sentence by itself), so it's followed with か to mark that it's a smaller hypothetical sentence in a longer sentence.
So かのは doesn't really make sense as a cluster of particles here, because の can't follow か (it's either nominalizing the verb or adding an explanitory tone, and in both cases would be following the verb) and は can't directly follow a question.
You can say Vか*と*は, which turns the question-within-a-sentence into a quotation with と, or just leave out any other particle altogether and say Vか、with a pause between it and the rest of the sentence.
In that case, I'm assuming (VのかVの) は would be grammatically correct, since Vの is a noun and NかNは seems to be a valid grammar pattern, but would be less natural to say, right?
I think I'd need an example of what you would try to say with that pattern, because I can't think of any reason that would be used instead of VかVか and my instinct is that it's unnatural. Making up hypothetical sentence structures to see of they're valid when you don't have a use case for them is just going to end up confusing you, so unless you have a specific thing you want to say that the grammar point at face-value can't do, it's not really a necessary road to go down.
